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1. Given data on (yi, xi) for i = 1, , n, consider the following least...
Due: Jan 22 ECN 702 Econometrics II 1. Given data on (x) for i,n, consider the...
Due: Jan 22 ECN 702 Econometrics II 1. Given data on (x) for i,n, consider the following least square problem for a simple linear regression. bobi We assume the four linear regression model assumptions dicussed in class hold. 6) Compute the partial derivatives of the objective function. (ii) Put the derived partial derivatives in (i) equal to zeros. Explain why the resulting equa- tions are called ‘normal equation. (Hint: two n-dimesional vectors (voi-i and (wi)-1 are normal(-orthogonal) if Σ(-1 uiv.-0.)...
Consider the linear model: Yi = α0 + α1(Xi − X̄) + ui. Find the OLS...
Consider the linear model: Yi = α0 + α1(Xi − X̄) + ui.
Find the OLS estimators of α0 and α1. Compare with the OLS
estimators of β0 and β1 in the standard model discussed in class
(Yi = β0 + β1Xi + ui).
Consider the linear model: Yį = ao + Q1(X; - X) + Ui. Find the OLS estimators of do and a1. Compare with the OLS estimators of Bo and B1 in the standard model discussed in...
Consider a simple linear regression model with nonstochastic regressor: Yi = β1 + β2Xi + ui....
Consider a simple linear regression model with nonstochastic regressor: Yi = β1 + β2Xi + ui. 1. [3 points] What are the assumptions of this model so that the OLS estimators are BLUE (best linear unbiased estimates)? 2. [4 points] Let βˆ and βˆ be the OLS estimators of β and β . Derive βˆ and βˆ. 12 1212 3. [2 points] Show that βˆ is an unbiased estimator of β .22
Consider the following regression model: Xi = Bo + Bixi + y; where yi is individual...
Consider the following regression model: Xi = Bo + Bixi + y; where yi is individual i's University GPA and xi is the individual's high school grades. a. What do you think is in ui? Do you think E[ulx) = 0? Suggest a variable that you think might affect University GPA that isn't included in the regression equation but should be. c. What sign of bias would you expect in an OLS regression of y on x? Briefly explain. d....
1. Suppose the true conditional mean function is but by mistake, a researcher ran least square...
1. Suppose the true conditional mean function is but by mistake, a researcher ran least square regression without the X term as in Assume cou (Xi, U)-0, E Xil]-o and E [x?]-: i. Is his/her estimate consistent for β? If not, show which OLS assumption fails and discuss potential solutions. 2. Assume the structural equation is where E [111x,-0. It was discovered that we observe Xi with a measurement error wi instead of the real value Xi It is known...
Consider the regression model given by: Yi = βo + β1Xi + β2Zi+ ui Suppose that...
Consider the regression model given by: Yi = βo + β1Xi + β2Zi+ ui Suppose that an econometrician wishes to test the null hypothesis given by: Ho: β1 + β2 = 0 Use this null hypothesis to specify a restricted form of the regression model (in a form that may be estimated using an OLS estimation procedure). State the equation that you could estimate as the restricted version of this model.
1. Suppose the true conditional mean function is but by mistake, a researcher ran least square...
1. Suppose the true conditional mean function is but by mistake, a researcher ran least square regression without the X term as in Assume cou (Xi , U) 0, E [Xn] O and E [x?-I . Is his/her estimate consistent for β,? If not, show which OLS assumption fails and discuss potential solutions. 2. Assume the structural equation is where E [u:|X0. It was discovered that we observe Xi with a measurement error wi instead of the real value X,...
Consider the simple linear regression model: Yi = Bo + Bilitei, i = 1,...,n. with the...
Consider the simple linear regression model: Yi = Bo + Bilitei, i = 1,...,n. with the least squares estimates ỘT = (Bo ß1). We observe a new value of the predictor: x] = (1 xo). Show that the expression for the 100(1 - a)% prediction interval reduces to the following: . (xo – x2 Ēo + @130 Etap 11+ntan (x; – 7)2
In the simple linear regression with zero-constant item for (xi , yi) where i = 1, 2, · · · , n, Yi = βxi + i where {i} n i=1 are i.i.d. N(0, σ2 ). (a) Derive the normal equation that the LS estimator...
In the simple linear regression with zero-constant item for (xi , yi) where i = 1, 2, · · · , n, Yi = βxi + i where {i} n i=1 are i.i.d. N(0, σ2 ). (a) Derive the normal equation that the LS estimator, βˆ, satisfies. (b) Show that the LS estimator of β is given by βˆ = Pn i=1 P xiYi n i=1 x 2 i . (c) Show that E(βˆ) = β, V ar(βˆ) = σ...
Exercise5 Consider a linear model with n -2m in which yi Bo Pi^i +ei,i-1,...,m, and Here €1, ,En are 1.1.d. from N(0,ơ)...
Exercise5 Consider a linear model with n -2m in which yi Bo Pi^i +ei,i-1,...,m, and Here €1, ,En are 1.1.d. from N(0,ơ), β-(A ,A, β), and σ2 are unknown parameters, zı, known constants with x1 +... + Xm-Tm+1 + +xn0 , zn are 1, write the model in vector form as Y = Xß+ε describing the entries in the matrix X. 2, Determine the least squares estimator β of β.
Exercise5 Consider a linear model with n -2m in which...
Due: Jan 22 ECN 702 Econometrics II 1. Given data on (x) for i,n, consider the following least square problem for a simple linear regression. bobi We assume the four linear regression model assumptions dicussed in class hold. 6) Compute the partial derivatives of the objective function. (ii) Put the derived partial derivatives in (i) equal to zeros. Explain why the resulting equa- tions are called ‘normal equation. (Hint: two n-dimesional vectors (voi-i and (wi)-1 are normal(-orthogonal) if Σ(-1 uiv.-0.)...
Consider the linear model: Yi = α0 + α1(Xi − X̄) + ui.
Find the OLS estimators of α0 and α1. Compare with the OLS
estimators of β0 and β1 in the standard model discussed in class
(Yi = β0 + β1Xi + ui).
Consider the linear model: Yį = ao + Q1(X; - X) + Ui. Find the OLS estimators of do and a1. Compare with the OLS estimators of Bo and B1 in the standard model discussed in...
Consider the following regression model: Xi = Bo + Bixi + y; where yi is individual i's University GPA and xi is the individual's high school grades. a. What do you think is in ui? Do you think E[ulx) = 0? Suggest a variable that you think might affect University GPA that isn't included in the regression equation but should be. c. What sign of bias would you expect in an OLS regression of y on x? Briefly explain. d....
1. Suppose the true conditional mean function is but by mistake, a researcher ran least square regression without the X term as in Assume cou (Xi, U)-0, E Xil]-o and E [x?]-: i. Is his/her estimate consistent for β? If not, show which OLS assumption fails and discuss potential solutions. 2. Assume the structural equation is where E [111x,-0. It was discovered that we observe Xi with a measurement error wi instead of the real value Xi It is known...
1. Suppose the true conditional mean function is but by mistake, a researcher ran least square regression without the X term as in Assume cou (Xi , U) 0, E [Xn] O and E [x?-I . Is his/her estimate consistent for β,? If not, show which OLS assumption fails and discuss potential solutions. 2. Assume the structural equation is where E [u:|X0. It was discovered that we observe Xi with a measurement error wi instead of the real value X,...
Consider the simple linear regression model: Yi = Bo + Bilitei, i = 1,...,n. with the least squares estimates ỘT = (Bo ß1). We observe a new value of the predictor: x] = (1 xo). Show that the expression for the 100(1 - a)% prediction interval reduces to the following: . (xo – x2 Ēo + @130 Etap 11+ntan (x; – 7)2
Exercise5 Consider a linear model with n -2m in which yi Bo Pi^i +ei,i-1,...,m, and Here €1, ,En are 1.1.d. from N(0,ơ), β-(A ,A, β), and σ2 are unknown parameters, zı, known constants with x1 +... + Xm-Tm+1 + +xn0 , zn are 1, write the model in vector form as Y = Xß+ε describing the entries in the matrix X. 2, Determine the least squares estimator β of β.
Exercise5 Consider a linear model with n -2m in which...
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